i shall use the usual notation for the sides of the triangle and the semi perimeter.AB = c, BC = a and AC = b and s = (1/2)*(a+b+c)now by the tangency property of the in circle we have AD = s-a and AM = b/2 as M is the midpoint of AC. now DM = AM - AD = b/2 - (s-a) = a/2 - c/2 = (1/2)*(a-c). this implies that DM = (1/2)*(BC - AB).Q. E. D.